On the use of a nascent delta function in radiative-transfer calculations for multi-layer media subject to Fresnel boundary and interface conditions

The “pre-processing” procedure and the “break-point” analysis developed in a previous work based on the ADO (analytical discrete ordinates) method are used, along with a nascent delta function to describe the polar-angle dependence of an incident beam, to solve the classical albedo problem for radiative transfer in a plane-parallel, multi-layer medium subject to Fresnel boundary and interface conditions. As a result of the use of a nascent delta function, rather than the Dirac distribution, to model the polar-angle dependence of the incident beam, the computational work is significantly simplified (since a particular solution is not required) in comparison to an approach where both the polar-angle and the azimuthal-angle dependence of the incident beam are formulated in terms of Dirac delta distributions. The numerical results from this approach are (when a sufficiently small “narrowness” parameter is used to define the nascent delta) found to be in complete agreement with already reported (high-quality) results for a set of challenging multi-layer problems.     

Some exact results basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

Six exact solutions (related to the conservation of number, energy and momentum) of the linearized Boltzmann equations for a binary mixture of rigid spheres, for the case of isotropic scattering in the center-of-mass system, are reported. The verification of the reported exact solutions (collisional invariants) is based on a recently reported explicit formulation of the linearized Boltzmann equation for a binary mixture of rigid spheres. Elementary analysis is used also to establish a basic flow condition.     

The McCormack model for gas mixtures: The temperature-jump problem

An analytical version of the discrete-ordinates method (the ADO method) is used to establish a concise and particularly accurate solution to the temperature-jump problem for a binary gas mixture described by the McCormack kinetic model. The solution yields, in addition to the temperature-jump coefficient for the general (specular–diffuse) case of Maxwell boundary conditions for each of the two species, the density and temperature profiles for both types of particles. Numerical results are reported for two binary mixtures (Ne-Ar and He-Xe) with various molar concentrations. The algorithm is considered especially easy to use, and the developed (FORTRAN) code requires typically less than a second on a 2.2 GHz Pentium 4 machine to compute all quantities of interest.     

An inverse source problem in radiative transfer

The spherical-harmonics method is used to develop a solution to an inverse source problem in radiative transfer. It is assumed that, with the exception of the inhomogeneous source term, all aspects of the radiation-transport problem are known, and we seek to determine the inhomogeneous source term from specified angular distributions of radiation exiting the two surfaces of a homogeneous plane-parallel medium. Anisotropic scattering is included in the monochromatic radiative-transfer model and general reflecting boundary conditions are considered.     

An analytical discrete-ordinates solution for dual-mode heat transfer in a cylinder

A modern analytical version of the discrete-ordinates method is used along with Hermite cubic splines and Newton's method to solve a class of coupled nonlinear radiation-conduction heat-transfer problems in a solid cylinder. Computational details of the solution are discussed, and the algorithm is implemented to establish high-quality results for various data sets which include some difficult cases.